let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom f & ex r being Real st rng f = {r} holds

( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0 ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom f & ex r being Real st rng f = {r} implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0 ) ) )

reconsider cf = REAL --> (In (0,REAL)) as Function of REAL,REAL ;

set R = cf;

set L = cf;

for p being Real holds cf . p = 0 * p ;

then reconsider L = cf as LinearFunc by Def3;

assume A7: Z c= dom f ; :: thesis: ( for r being Real holds not rng f = {r} or ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0 ) ) )

given r being Real such that A8: rng f = {r} ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0 ) )

(f `| Z) . x = 0

let x0 be Real; :: thesis: ( x0 in Z implies (f `| Z) . x0 = 0 )

assume A15: x0 in Z ; :: thesis: (f `| Z) . x0 = 0

then A16: f is_differentiable_in x0 by A10;

then ex N being Neighbourhood of x0 st

( N c= dom f & ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) ) ;

then consider N being Neighbourhood of x0 such that

A17: N c= dom f ;

A18: for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

thus (f `| Z) . x0 = diff (f,x0) by A14, A15, Def7

.= L . j by A16, A17, A18, Def5

.= 0 ; :: thesis: verum

( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0 ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom f & ex r being Real st rng f = {r} implies ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0 ) ) )

reconsider cf = REAL --> (In (0,REAL)) as Function of REAL,REAL ;

set R = cf;

now :: thesis: for h being non-zero 0 -convergent Real_Sequence holds

( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0 )

then reconsider R = cf as RestFunc by Def2;( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0 )

let h be non-zero 0 -convergent Real_Sequence; :: thesis: ( (h ") (#) (cf /* h) is convergent & lim ((h ") (#) (cf /* h)) = 0 )

hence (h ") (#) (cf /* h) is convergent ; :: thesis: lim ((h ") (#) (cf /* h)) = 0

((h ") (#) (cf /* h)) . 0 = 0 by A2;

hence lim ((h ") (#) (cf /* h)) = 0 by A6, SEQ_4:25; :: thesis: verum

end;A2: now :: thesis: for n being Nat holds ((h ") (#) (cf /* h)) . n = 0

then A6:
(h ") (#) (cf /* h) is V8()
by VALUED_0:def 18;let n be Nat; :: thesis: ((h ") (#) (cf /* h)) . n = 0

A4: rng h c= dom cf ;

A5: n in NAT by ORDINAL1:def 12;

thus ((h ") (#) (cf /* h)) . n = ((h ") . n) * ((cf /* h) . n) by SEQ_1:8

.= ((h ") . n) * (cf . (h . n)) by A5, A4, FUNCT_2:108

.= ((h ") . n) * 0

.= 0 ; :: thesis: verum

end;A4: rng h c= dom cf ;

A5: n in NAT by ORDINAL1:def 12;

thus ((h ") (#) (cf /* h)) . n = ((h ") . n) * ((cf /* h) . n) by SEQ_1:8

.= ((h ") . n) * (cf . (h . n)) by A5, A4, FUNCT_2:108

.= ((h ") . n) * 0

.= 0 ; :: thesis: verum

hence (h ") (#) (cf /* h) is convergent ; :: thesis: lim ((h ") (#) (cf /* h)) = 0

((h ") (#) (cf /* h)) . 0 = 0 by A2;

hence lim ((h ") (#) (cf /* h)) = 0 by A6, SEQ_4:25; :: thesis: verum

set L = cf;

for p being Real holds cf . p = 0 * p ;

then reconsider L = cf as LinearFunc by Def3;

assume A7: Z c= dom f ; :: thesis: ( for r being Real holds not rng f = {r} or ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0 ) ) )

given r being Real such that A8: rng f = {r} ; :: thesis: ( f is_differentiable_on Z & ( for x being Real st x in Z holds

(f `| Z) . x = 0 ) )

A9: now :: thesis: for x0 being Real st x0 in dom f holds

f . x0 = r

f . x0 = r

let x0 be Real; :: thesis: ( x0 in dom f implies f . x0 = r )

assume x0 in dom f ; :: thesis: f . x0 = r

then f . x0 in {r} by A8, FUNCT_1:def 3;

hence f . x0 = r by TARSKI:def 1; :: thesis: verum

end;assume x0 in dom f ; :: thesis: f . x0 = r

then f . x0 in {r} by A8, FUNCT_1:def 3;

hence f . x0 = r by TARSKI:def 1; :: thesis: verum

A10: now :: thesis: for x0 being Real st x0 in Z holds

f is_differentiable_in x0

hence A14:
f is_differentiable_on Z
by A7, Th9; :: thesis: for x being Real st x in Z holds f is_differentiable_in x0

let x0 be Real; :: thesis: ( x0 in Z implies f is_differentiable_in x0 )

assume A11: x0 in Z ; :: thesis: f is_differentiable_in x0

then consider N being Neighbourhood of x0 such that

A12: N c= Z by RCOMP_1:18;

A13: N c= dom f by A7, A12;

for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

end;assume A11: x0 in Z ; :: thesis: f is_differentiable_in x0

then consider N being Neighbourhood of x0 such that

A12: N c= Z by RCOMP_1:18;

A13: N c= dom f by A7, A12;

for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

proof

hence
f is_differentiable_in x0
by A13; :: thesis: verum
let x be Real; :: thesis: ( x in N implies (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )

reconsider xx = x, xx0 = x0 as Element of REAL by XREAL_0:def 1;

assume x in N ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

hence (f . x) - (f . x0) = r - (f . x0) by A9, A13

.= r - r by A7, A9, A11

.= (L . (xx - xx0)) + 0

.= (L . (x - x0)) + (R . (x - x0)) ;

:: thesis: verum

end;reconsider xx = x, xx0 = x0 as Element of REAL by XREAL_0:def 1;

assume x in N ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

hence (f . x) - (f . x0) = r - (f . x0) by A9, A13

.= r - r by A7, A9, A11

.= (L . (xx - xx0)) + 0

.= (L . (x - x0)) + (R . (x - x0)) ;

:: thesis: verum

(f `| Z) . x = 0

let x0 be Real; :: thesis: ( x0 in Z implies (f `| Z) . x0 = 0 )

assume A15: x0 in Z ; :: thesis: (f `| Z) . x0 = 0

then A16: f is_differentiable_in x0 by A10;

then ex N being Neighbourhood of x0 st

( N c= dom f & ex L being LinearFunc ex R being RestFunc st

for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) ) ;

then consider N being Neighbourhood of x0 such that

A17: N c= dom f ;

A18: for x being Real st x in N holds

(f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

proof

reconsider j = 1 as Element of REAL by XREAL_0:def 1;
let x be Real; :: thesis: ( x in N implies (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0)) )

reconsider xx = x, xx0 = x0 as Element of REAL by XREAL_0:def 1;

assume x in N ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

hence (f . x) - (f . x0) = r - (f . x0) by A9, A17

.= r - r by A7, A9, A15

.= (L . (xx - xx0)) + 0

.= (L . (x - x0)) + (R . (x - x0)) ;

:: thesis: verum

end;reconsider xx = x, xx0 = x0 as Element of REAL by XREAL_0:def 1;

assume x in N ; :: thesis: (f . x) - (f . x0) = (L . (x - x0)) + (R . (x - x0))

hence (f . x) - (f . x0) = r - (f . x0) by A9, A17

.= r - r by A7, A9, A15

.= (L . (xx - xx0)) + 0

.= (L . (x - x0)) + (R . (x - x0)) ;

:: thesis: verum

thus (f `| Z) . x0 = diff (f,x0) by A14, A15, Def7

.= L . j by A16, A17, A18, Def5

.= 0 ; :: thesis: verum